Extension card · Rough
Interval rough CODAS (Cherif and Frikha, 2021)
Interval rough CODAS is the form of CODAS used when several experts score the same criterion as an interval and the disagreement between them needs to be preserved. It directly supports a group decision and still ranks the result with a single assessment score.
Base method
CODAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Rough →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Four things change. The two-distance comparison logic does not.
Cells. In crisp CODAS every cell is a single number. Here every cell is an interval rough number: a lower approximation interval and an upper approximation interval, held together, in order, as four numbers (a ≤ b ≤ c ≤ d). These bounds are not entered by hand; they are derived beforehand from several experts' interval judgements using the rough-number rule explained on the data-type card. Criterion weights are also reduced to a single group weight by averaging across experts. The method directly supports a group decision.
Scale equalisation. Crisp CODAS divides every cell by the column's largest value, or by the ratio to its smallest. Here each component is divided by its own cross-conjugate. On a benefit criterion, the smallest component (a) is set against the column's largest d, and the largest component (d) against the column's smallest a. Components b and c are cross-divided against each other in the same way. On a cost criterion this cross-division runs in the opposite direction. This cross-pairing is designed to keep the ordering (a ≤ b ≤ c ≤ d) intact after normalisation too.
Distance. In crisp CODAS the negative-ideal is built from the column's worst value. Here a negative-ideal interval rough number is built by taking the smallest value separately for each of the four components. Euclidean and Taxicab distances come from the average of the differences computed across the four components.
Result and defuzzification. The assessment score is built with the same pairwise comparison rule as crisp CODAS. There is no separate defuzzification step; the four components only merge into a single number during the distance calculation.
DecisionMind fixes the threshold value at τ=0.03, the value the source itself uses, for classical interval rough CODAS. This differs from the 0.02 used in most of the other CODAS extensions in the family.
How to Read the Output
The assessment score is read in the same way as in crisp CODAS. It is a relative positional measure, not a percentage, and cannot be compared with a different analysis. The difference is here: because the score comes from a four-component interval, it is a composite of both disagreement between experts and each expert's own interval width. The two layers merge into the same number.
Thus instead of writing:
"The interval rough CODAS score shows only the disagreement between experts"
the report should read:
"The score carries both the disagreement between experts and each expert's own interval width together; unless the two are reported separately, it is not visible which one is driving the result"
When to Prefer This over the Base Method
Use this extension when several experts score the same criterion as an interval, and both the disagreement between experts and the within-expert interval width need to be preserved together. If there is only one expert, or experts give a single number, a rough structure cannot be built; the rule on the data-type card applies here too. Measured criteria should not be brought into this extension. If the matrix must be of a single type, a measured value is written as a single interval rough number in which all four components are equal. Base CODAS's exit condition applies here exactly as it does there.
Mistakes Specific to This Extension
Failing to notice that the ordering (a ≤ b ≤ c ≤ d) breaks down after normalisation. On a cost criterion, cross-division can break this ordering because the divisors differ. The engine does not re-check this ordering after normalisation; the user must check it.
Using inter-expert weights directly without averaging them. The method expects weights to have already been averaged across the group beforehand. Giving a single expert's weight as if it were the group weight produces a wrong result.
Marking a criterion's direction incorrectly. In the illustrative example below, treating a benefit criterion as a cost swaps the two middle-ranked risks.
Accepting a number from a published paper's printed intermediate table as correct without question. As shown in the case below, values the engine recomputes with its own formula can diverge slightly from the table the source prints.
The governing principle is this:
The four components of an interval rough number carry both within-expert and between-expert uncertainty. Their ordering must not break down after normalisation, weights must be averaged across the group beforehand, and a printed table's figure must never be accepted without question.
Cases
The first case is a genuine literature case: Cherif and Frikha's (2021) H₂S risk-assessment example for the Sfax Hannibal gas facility. The second case is an illustrative construction.
1. Industrial safety: Assessing five risk types at a gas facility (Cherif and Frikha, 2021)
Five risk types at a gas processing facility (explosion, fire, leakage, respiratory fatigue, control-instrument malfunction) were assessed against four criteria: safety benefit (higher is better), exposure frequency (lower is better), severity (lower is better), and environmental impact (lower is better). Three experts arrived at interval rough numbers from judgements given on a verbal scale, and also averaged their weights among themselves. The threshold value is τ=0.03.
| Risk | Safety benefit | Exposure frequency (cost) | Severity (cost) | Environmental impact (cost) |
|---|---|---|---|---|
| Explosion | [5.89; 6.78][7.22; 8.11] | [6.00; 8.00][7.89; 8.78] | [1.00; 1.00][1.22; 2.11] | [1.22; 2.11][1.89; 2.78] |
| Fire | [7.45; 8.56][9.00; 9.00] | [7.22; 8.11][7.89; 8.78] | [1.22; 2.11][1.89; 2.78] | [1.22; 2.11][3.22; 4.11] |
| Leakage | [7.22; 8.11][9.00; 9.00] | [1.89; 2.78][4.00; 6.00] | [1.00; 1.00][3.22; 4.11] | [2.22; 4.56][5.00; 5.00] |
| Respiratory fatigue | [5.22; 6.11][7.22; 8.10] | [3.89; 4.78][5.22; 6.11] | [3.00; 3.00][3.89; 4.78] | [7.89; 8.78][9.00; 9.00] |
| Malfunction | [5.22; 6.11][7.22; 8.11] | [1.22; 2.11][3.89; 4.78] | [1.22; 2.11][2.00; 4.00] | [5.89; 6.78][7.22; 8.11] |
| Weight | 0.393 | 0.103 | 0.323 | 0.180 |
The method normalises every component against its own cross-conjugate, multiplies by the weight, builds the negative-ideal from the smallest of each component, and computes the Euclidean and Taxicab distance across the four components.
| Risk | Assessment score | Rank |
|---|---|---|
| Explosion | 1.2114 | 1 |
| Leakage | 0.8623 | 2 |
| Fire | 0.3497 | 3 |
| Malfunction | -0.0389 | 4 |
| Respiratory fatigue | -2.3846 | 5 |
The result reads as follows. Explosion sits at a relatively middling position on the safety-benefit criterion. It has, however, the lowest, that is the best, intervals on the severity and environmental-impact criteria; these two criteria's combined weight (0.503) exceeds the weight of safety benefit (0.393). Respiratory fatigue comes last because it has the highest, that is worst, interval on the environmental-impact criterion. The source article's own printed Table 9 gives a different order for Fire and Leakage (fire second, leakage third) from the order recomputed with the engine's own formula (leakage second, fire third). This difference could not be reproduced under any combination of criterion directions and most likely stems from the source's own intermediate rounding; the engine is internally consistent with its own formula.
To see how robust the decision is, the weights were changed and the calculation rerun. The safety-benefit weight was raised from 0.393 to 0.57, the severity weight lowered from 0.323 to 0.146, and the other two weights held fixed. Under this change Fire moves ahead (0.6932) and Explosion drops to second (0.6657). This means that Explosion's first place is sensitive to the weight on safety benefit.
In the report: "With the given weights (safety benefit 0.393, exposure 0.103, severity 0.323, environmental impact 0.18) and a threshold of τ=0.03, Explosion has the highest assessment score (1.21); once the safety-benefit weight is raised to about 0.57, Fire moves ahead (0.6932 / 0.6657). There is a small discrepancy against the source article's printed ranking, and this discrepancy is separately documented."
Source: Cherif and Frikha (2021), Table 5 (group interval rough decision matrix), Table 7 (averaged weights), Table 9 (published result). The assessment scores were obtained by independently rerunning DecisionMind's interval rough CODAS engine on this matrix. The engine's own ranking (Explosion, Leakage, Fire, Malfunction, Respiratory fatigue) diverges from the article's printed ranking (Explosion, Fire, Malfunction, Leakage, Respiratory fatigue) at second and third place; the detail is in the verification notes.
2. Assessment and testing: An examination centre's choice of digital exam-software provider
A national examination centre will contract with one of three providers for the digital exam software to be used in its upcoming sessions. The criteria are: level of data security, system downtime risk (lower is better) and licence cost (lower is better). Five officials at the centre have scored each provider as an interval, and interval rough numbers have been derived from these scores.
The method normalises the three providers' interval rough numbers, multiplies by the weights, and builds the negative-ideal from the provider with the lowest-component interval. Suppose the most secure provider also has the highest licence cost. It nonetheless comes first, because the weight on the data-security criterion exceeds that on cost.
The centre's hesitation is this. The gap between the five officials' intervals is narrow for the most secure provider but wide for the second provider. This width gap shows that the officials disagree about the second provider, and it should be reported separately alongside the score.
In the report: "With the high weight given to data security, the most secure provider reaches the highest assessment score; the interval width for the second provider shows disagreement among the officials, and this should be tracked separately."
3. What Not to Do
Had safety benefit been mistakenly counted as a cost criterion in the illustrative table, that is, had the cross-normalisation been applied in the reverse direction, third-ranked Fire and fourth-ranked Malfunction would swap places (Malfunction 0.182, Fire -0.163); in the correct ranking Fire is third and Malfunction fourth. The second error is taking the widest interval given by any one of the five officials as "the group view" without averaging the intervals they gave separately; this skips the rough number's own calculation rule. The third error is accepting the source article's printed Table 9 ranking unquestioningly as "the correct ranking" and treating the engine's own formula-derived, differing ranking as wrong; which one is affected by intermediate rounding needs separate examination.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/ir-codas
Cherif, M. R., & Frikha, H. M. (2021). An extension of the CODAS method based on interval rough numbers for multi-criteria group decision making. Multiple Criteria Decision Making, 16, 25–43. DOI: 10.22367/mcdm.2021.16.02
Keshavarz Ghorabaee, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2016). A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Economic Computation and Economic Cybernetics Studies and Research, 50(3), 25–44. (no DOI. This article is not registered on Crossref; see the Sources section of the base CODAS card.)
Pamucar, D., Stevic, Z., & Zavadskas, E. K. (2018). Integration of interval rough AHP and interval rough MABAC methods for evaluating university web pages. Applied Soft Computing, 67, 141–163. DOI: 10.1016/j.asoc.2018.02.057
Pawlak, Z. (1982). Rough sets. International Journal of Computer & Information Sciences, 11(5), 341–356. DOI: 10.1007/BF01001956